Mike Dougherty wrote:
On 10/5/07, Richard Loosemore <[EMAIL PROTECTED]> wrote:
I hear you, but let me quickly summarize the reason why I introduced GoL
as an example.
Thank you. I appreciate the confirmation of understanding my point.
I have observed many cases where the back and forth bickering over
email lists have been based in an unwillingness to concede an other's
point. I am the first to admit that I have more questions than
answers.
I wanted to use GoL as a nice-and-simple example of a system whose
overall behavior (in this case, the existence of certain patterns that
are "stable" or "interesting") seems impossible to predict from a
knowledge of the rules. I only wanted to use GoL to *illustrate* the
general class, not because I was interested in GoL per se.
Gotcha - GoL is an example case of a class. You threw it out there to
make a point. Let's just say is the only symbol on the table. In
order to assimilate the idea you are proposing, the model needs to be
examined. So if we discuss this one example it is not to the
exclusion of the concept you're trying to illustrate, but a precursor
to it. In my own concept formation, this step is like including
libraries or compiling a function. I think sometimes you get
frustrated that it takes so long for people accomplish this step.
Part of the problem is that email is such a low bandwidth medium.
(another part is that the smarter we are, the quicker we "get" stuff
and we assume others should be as capable)
The important thing is that this idea (that there are some systems that
show interesting, but unexplainable, behavior at the global level) has
much greater depth and impact than people have previously thought.
Can you give an example of a ruleset that CAN be used to predict
global behavior?
"interesting but unexplainable behavior" - would you define this class
to include chaos or chaotic systems? I'm trying to reason to the
general case, but I don't have enough other properties of the class in
mind to usefully visualize. (conceptualize?) I think those
researchers who have invested in studying chaos are people who have
given this idea a great deal of depth and impact. It's a hard problem
because our normal 'scientific' method fails almost by definition. I
believe the framework you have discussed is a proposal for a method of
investigating this behavior. Am I far off, or am I in the general
vicinity?
Thanks for this (what a relief to just communicate with someone in a
relaxed way!).
About discussing GoL itself as the first example of the class, that's
fine, but some of the simplicity of GoL can make it misleading -- there
are so many things that have been said about it that we can easily get
distracted by those. (I am beginning to realize, now, that although it
is a memorable example, these side effects have made it a pain to use
for my example. It is not that it is not a good example, just that it
has so much baggage).
So, fire away with any questions about how GoL relates, and I'll try to
say how they fit with what I was trying to say.
About your second question "Can you give an example of a ruleset that
CAN be used to predict global behavior?", well, the short answer is that
you can choose any scientific explanation you want. "Ruleset" must be
understood as meaning "low level equations or mechanisms that drive the
system".
My stock example: planetary motion. Newton (actually Tycho Brahe,
Kepler, et al) observed some global behavior in this system: the orbits
are elliptical and motion follows Kepler's other laws. This corresponds
to someone seeing Game of Life for the first time, without knowing how
it works, and observing that the motion is not purely random, but seems
to have some regular patterns in it.
Having noticed the global regularities, the next step, for Newton, was
to try to find a compact explanation for them. He was looking for the
underlying rules, the low-level mechanisms. He eventually realised (a
long story of course!) that an inverse square law of gravitation would
predict all of the behavior of these planets. This corresponds to a
hypothetical case in which a person seeing those Game of Life patterns
would somehow deduce that the rules that must be giving rise to the
patterns are the particular rules that appear in GoL. And, to be
convincing, they would have to prove that the rules gave rise to the
behavior.
(Caveat: we have to fuzz the analogy somewhat and say that the observer
cannot simply look at the behavior of individual cells, but only see a
hazy picture of the larger scale structures ... if they could see every
cell clearly they could reason about the rules and deduce them. This is
a weakness in the analogy, but I am sure you will be able to imagine
other circumstances in which, for some reason, the lowest level
mechanisms are not nakedly apparent).
(To make this issue really stand out clearly, imagine that I pulled out
a sheet of paper, drew a whole bunch of patterns, togther with their
exact periods, that I wanted to see as the global behavior of a GoL-like
cellular automaton, and I said "Find me some rules that will generate
this set of patterns!". Would it be even slightly feasible for someone
to take on such a challenge?)
So: the question is, if observers could see only (say) snapshots of the
patterns in GoL that are "stable" or "interesting" or "recurring", COULD
they actually do what Newton did and figure out the underlying rules?
or, vice versa, if they had access to the rules, could they PROVE that
those rules generate the patterns? Could they predict the existence of
those patterns?
We have good reason to believe, after studying systems like GoL, that
even if there exists a compact theory that would let us predict the
patterns from the rules (equivalent to predicting planetary dynamics
given the inverse square law of gravitation), such a theory is going to
be so hard to discover that we may as well give up and say that it is a
waste of time trying. Heck, maybe it does exist, but that's not the
point: the point is that there appears to be little practical chance of
finding it.
Now, this is only just true of GoL, but GoL is so simple that we might
almost be tempted to have a go at finding a theory. But what is just
about true of GoL is true in spades of all the other, vastly more
complicated systems that exist. Imagine a cellular automaton in which
the rules that govern the cells have to be written down with an
algorithm that is not three or four lines long, but a million lines
long. If we saw patterns in that system, would we feel confident that a
compact theory does exist, or that we could find it?
What we have to do is concede that there are some systems in which the
components of the system interact in the most horribly nonlinear and
tangled way, and that those systems might show interesting regularities
in their global behavior (something equivalent to the patterns in GoL,
or the motion of the planets), BUT we have absolutely no practical hope
of finding a compact theory to explain those global regularities.
I'll stop at this point in the argument.
One final quick word about "chaotic" systems.
Technically, these do fall into the class of complex systems, but they
are a kind of special case. Different people describe the relationship
in different ways, but one way is to say that they are such "simple"
complex systems that they allow something compact to be said about their
global behavior. And the "something" can actually be said in
mathematical form, so what has happened (historically) is that
mathematicians have seized on chaotic systems and done huge amounts of
analysis to find out their properties. At the beginning, mathematicians
were faced with a big class of complex systems as possible objects of
study, but they looked at them, noticed a diabolically simple subset
over in the corner, and said "Well, we know how to do a certain amount
of work on *these* critters, so let's ignore all the other stuff and
just concentrate on the ones that we can get a handle on". So chaos
theory was born. It's fun, but it does not say anything about the
general class of complex systems.
Let me know if this makes sense.
Richard Loosemore
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